Topological phases of matter, like fractional quantum Hall systems, can host anyon excitations with fractional electric charge. More generally, when topological phases with anyons have global symmetries, the anyons can carry fractional quantum numbers under those symmetries. Remarkably, some choices of fractional quantum numbers are anomalous, that is, they are physically allowed to exist, but only when the (2+1)D system lives on the surface of a bulk (3+1)D symmetry-protected topological phase like a topological insulator. Given abstract algebraic data specifying these quantum numbers, I will explain how to determine whether the resulting (2+1)D theory is anomalous and, if so, how to describe the required bulk theory.

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